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Integral Cayley graphs and groups
We solve two open problems regarding the classification of certain classes of
Cayley graphs with integer eigenvalues. We first classify all finite groups
that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a
problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups
was introduced by Klotz and Sander. These are groups for which every Cayley
graph has only integer eigenvalues. In the second part of the paper, all Cayley
integral groups are determined.Comment: Submitted June 18 to SIAM J. Discrete Mat
Groups all of whose undirected Cayley graphs are integral
Let be a finite group, be a set such that if
, then , where denotes the identity element of .
The undirected Cayley graph of over the set is the graph
whose vertex set is and two vertices and are adjacent whenever
. The adjacency spectrum of a graph is the multiset of all
eigenvalues of the adjacency matrix of the graph. A graph is called integral
whenever all adjacency spectrum elements are integers. Following Klotz and
Sander, we call a group Cayley integral whenever all undirected Cayley
graphs over are integral. Finite abelian Cayley integral groups are
classified by Klotz and Sander as finite abelian groups of exponent dividing
or . Klotz and Sander have proposed the determination of all non-abelian
Cayley integral groups. In this paper we complete the classification of finite
Cayley integral groups by proving that finite non-abelian Cayley integral
groups are the symmetric group of degree , and
for some integer , where is the
quaternion group of order .Comment: Title is change
Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory
The real monomial representations of Clifford algebras give rise to two
sequences of bent functions. For each of these sequences, the corresponding
Cayley graphs are strongly regular graphs, and the corresponding sequences of
strongly regular graph parameters coincide. Even so, the corresponding graphs
in the two sequences are not isomorphic, except in the first 3 cases. The proof
of this non-isomorphism is a simple consequence of a theorem of Radon.Comment: 13 pages. Addressed one reviewer's questions in the Discussion
section, including more references. Resubmitted to JACODES Math, with updated
affiliation (I am now an Honorary Fellow of the University of Melbourne
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